Signal and System Approximation from General Measurements

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley-Wiener space PW_\pi^1. We consider approximation processes, where the input signal is not directly used to generate the system output, but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday" in the Applied and Numerical Harmonic Analysis Series, Birkhauser (Springer-Verlag). Parts of this work have been presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing 2014 (ICASSP 2014

Sufficient stability bounds for slowly varying direct-form recursive linear filters and their applications in adaptive IIR filters

Journal ArticleAbstract-This correspondence derives a sufficient time-varying bound on the maximum variation of the coefficients of an exponentially stable time-varying direct-form homogeneous linear recursive filter. The stability bound is less conservative than all previously derived bounds for time-varying IIR systems. The bound is then applied to control the step size of output-error adaptive IIR filters to achieve bounded-input bounded-output (BIBO) stability of the adaptive filter. Experimental results that demonstrate the good stability characteristics of the resulting algorithms are included. This correspondence also contains comparisons with other competing output-error adaptive IIR filters. The results indicate that the stabilized method possesses better convergence behavior than other competing techniques

On stability of linear dynamic systems with hysteresis feedback

Abstract. The stability of linear dynamic systems with hysteresis in feedback is considered. While the absolute stability for memoryless nonlinearities (known as Lure’s problem) can be proved by the well-known circle criterion, the multivalued rate-independent hysteresis poses significant challenges for feedback systems, especially for proof of convergence to an equilibrium state correspondingly set. The dissipative behavior of clockwise input-output hysteresis is considered with two boundary cases of energy losses at reversal cycles. For upper boundary cases of maximal (parallelogram shape) hysteresis loop, an equivalent transformation of the closed-loop system is provided. This allows for the application of the circle criterion of absolute stability. Invariant sets as a consequence of hysteresis are discussed. Several numerical examples are demonstrated, including a feedback-controlled double-mass harmonic oscillator with hysteresis and one stable and one unstable poles configuration.acceptedVersio

Stability problems in constrained pendulum systems and time-delayed systems

In this dissertation, we study the boundary of stability of a class of linear mechanical systems as a function of a parameter. We consider two different systems under this class: a constrained double pendulum connected by a rigid rod and a state-feedback-controlled mechanical system with time delay. In the first system, the destabilizing parameter is the distance between the supports of the two pendulums. In the second system, the destabilizing parameter is the time delay. In the constrained double pendulum system, linear perturbation analysis is used to determine the natural frequency of the system. Our analysis reveals a zone of instability in what seemingly is an inherently stable configuration. This paradoxical behavior, which is not mentioned in the literature until now, is explained and a simple experiment confirms the instability predicted by the analysis. The approach is extended to a chain of pendulums consisting of n masses and n+1 links, which is a lumped parameter model for small vibrations of a catenary. Our work confirms the existence of asymmetric stable equilibrium configurations for a symmetric system. The problem of determining the critical distance for instability between two supporting points of a catenary has potential application in the design of novel mechanical switches, sensors, and valves. In the second part of the dissertation, we consider a linear mechanical system where a time delay exists in the linear state feedback control input. We seek a closed-form solution for the problem of determining the critical time delay for instability of the closed-loop system. Such a closed-form solution, which to the best of our knowledge is inexistent in the literature, offers an exact value for the critical time delay whereas a numerical solution is only approximate. We show that in the single-input/multi-output (SIMO) case of the class of systems under consideration, the problem may be reduced by using singular value decomposition to that of finding the roots of a certain polynomial. The obtained closed-form solution accurately predicts the smallest time delay that would render the SIMO system unstable when the control gain matrices have a unit rank. This technique however cannot be extended to the multi-input/multi-output case. Two numerical methods are therefore developed to solve this case. One method involves Newton’s iterations and the other method involves bisection for multiple functions

Independent estimation of input and measurement delays for a hybrid vertical spring-mass-damper via harmonic transfer functions

System identification of rhythmic locomotor systems is challenging due to the time-varying nature of their dynamics. Even though important aspects of these systems can be captured via explicit mechanics-based models, it is unclear how accurate such models can be while still being analytically tractable. An alternative approach for rhythmic locomotor systems is the use of data-driven system identification in the frequency domain via harmonic transfer functions (HTFs). To this end, the input-output dynamics of a locomotor behavior can be linearized around a stable limit cycle, yielding a linear, time-periodic system. However, few if any model-based or data-driven identification methods for time-periodic systems address the problem of input and measurement delays in the system. In this paper, we focus on data-driven system identification for a simple mechanical system and analyze its dynamics in the presence of input and measurement delays using HTFs. By exploiting the way input delays are modulated by the periodic dynamics, our results enable the separate, independent estimation of input and measurement delays, which would be indistinguishable were the system linear and time invariant. © 2015, IFAG

Performance analysis of switching systems

Performance analysis is an important aspect in the design of dynamic (control) systems. Without a proper analysis of the behavior of a system, it is impossible to guarantee that a certain design satisfies the system’s requirements. For linear time-invariant systems, accurate performance analyses are relatively easy to make and as a result also many linear (controller) design methods have appeared in the past. For nonlinear systems, on the other hand, such accurate performance analyses and controller design methods are in general not available. A main reason hereof is that nonlinear systems, as opposed to linear time-invariant systems, can have multiple steady-state solutions. Due to the coexistence of multiple steady-state solutions, it is much harder to define an accurate performance index. Some nonlinear systems, i.e. the so-called convergent nonlinear systems, however, are characterized by a unique steady-state solution. This steady-state solution may depend on the system’s input signals (e.g. reference signals), but is independent of the initial conditions of the system. In the past, the notion of convergent systems has already been proven to be very useful in the performance analysis of nonlinear systems with inputs. In this thesis, new results in the field of performance analysis of nonlinear systems with inputs are presented, based on the notion of convergent systems. One part of the thesis is concerned with the question "how to analyse the performance for a convergent system?" Since the behavior of a convergent system is independent of the initial conditions (after some transient time), simulation can be used to find the unique steady-state solution that corresponds to a certain input signal, but this can be very time-consuming. In this thesis, a computationally more efficient approach is presented to estimate the steady-state performance of harmonically forced Lur’e systems, in terms of nonlinear frequency response functions (nFRFs). This approach is based on the method of harmonic linearization. It provides both a linear approximation of the nFRF and an upper bound on the error between this linear approximation and the true nFRF. It is shown in several examples that the approximation of the nFRF is accurate, and that it provides more detailed information on the considered system than the often used ‘L2 gain’ performance index. An additional observation that is made, is that the method of harmonic linearization can sometimes be ‘misleading’ for Lur’e systems with a saturation-like nonlinearity: for the case that the harmonic balance equation has a unique solution, it is shown that the corresponding nonlinear system can have multiple distinct steady-state solutions. Another part of the thesis is concerned with the question "under what conditions is a system with inputs guaranteed to be convergent?" In particular two types of systems were investigated: switched linear systems and Lur’e systems with a saturation nonlinearity and marginally stable linear part. For the switched linear systems, it is assumed that the dynamics of all the separate linear modes are given. For this setting, it was investigated if it is possible to find a switching rule (which defines when to switch between the available modes) such that the closed-loop system is convergent. Both a state-based, an observer-based, and a time-based switching rule are presented that guarantee a convergent system, assuming some conditions on the linear dynamics are met. The second type of systems that are discussed, are Lur’e systems with a saturation nonlinearity and marginally stable linear part. For this type of systems, the goal was to find sufficient conditions under which the closed-loop system is convergent. Because of the marginally stable linear part, however, a quadratically convergent system cannot be obtained. Instead, sufficient conditions are discussed that guarantee uniform convergency of the system. The obtained theory is shown to be also applicable to a class of anti-windup systems with a marginally stable plant. For this class of systems, the results of the convergency-based performance analysis are compared with the analysis results of existing anti-windup methods. It is shown that the convergency-based performance analysis can in some cases provide more detailed information on the steady-state behavior of the system. The results of uniform convergency for anti-windup systems are shown to be also applicable in the field of production and inventory control of discrete-event manufacturing systems. Since a manufacturing machine has a certain production capacity and cannot produce at a negative rate, it can be seen as an integrator plant (input: production rate, output: amount of finished products) preceded by a saturation function. For this marginally stable plant, a controller was constructed in such a way that the closed-loop system is uniformly convergent. The controller was also implemented in the discrete-event domain and the results from discrete-event simulations were compared with those of continuous-time simulations. Similarly, the controller was also applied for the production and inventory control of a line of four manufacturing machines. For both the single machine and the line of four machines, the resulting controlled discrete-event systems are shown to have the desired tracking properties. Besides these theoretical and numerical results, also experimental results are presented in this thesis. By means of an electromechanical construction, several experimental results were obtained, and used to validate the theoretical results for both the switched linear systems and the anti-windup systems

Causal and Stable Input/Output Structures on Multidimensional Behaviours

In this work we study multidimensional (nD) linear differential behaviours with a distinguished independent variable called "time". We define in a natural way causality and stability on input/output structures with respect to this distinguished direction. We make an extension of some results in the theory of partial differential equations, demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the Pc operator defining the behaviour (Bc)0,y We also quote results which in effect characterise time autonomy for the general systems case. Stability is likewise characterized by a property of the transfer matrix. We prove this result for the 2D case and for the case of a single equation; for the general case it requires solution of an open problem concerning the geometry of a particular set in Cn. In order to characterize input/output stability we also develop new results on inclusions of kernels, freeness of variables, and closure with respect to S,S' and associated spaces, which are of independent interest. We also discuss stability of autonomous behaviours, which we beleive to be governed by a corresponding condition

Model Reduction for Aperiodically Sampled Data Systems

Two approaches to moment matching based model reduction of aperiodically sampled data systems are given. The term "aperiodic sampling" is used in the paper to indicate that the time between two consecutive sampling instants can take its value from a pre-specified finite set of allowed sampling intervals. Such systems can be represented by discrete-time linear switched (LS) state space (SS) models. One of the approaches investigated in the paper is to apply model reduction by moment matching on the linear time-invariant (LTI) plant model, then compare the responses of the LS SS models acquired from the original and reduced order LTI plants. The second approach is to apply a moment matching based model reduction method on the LS SS model acquired from the original LTI plant, and then compare the responses of the original and reduced LS SS models. It is proven that for both methods, as long as the original LTI plant is stable, the resulting reduced order LS SS model of the sampled data system is quadratically stable. The results from two approaches are compared with numerical examples